We assume ZF beamformers through the rest of this section to simplify the presentation. In general, optimized beamformers are preferred especially at the finite-SNR regime [1]}. For the problem considered in this paper, the optimized multicast beamformers can be obtained by solving the following weighted max-min optimization
\(\begin{array} {l}\underset{\begin{array}{c}\lbrace {\mathbf {v}}_{{\mathcal {U}}} \rbrace \end{array}}{\max } \underset{k \in \hat{{\mathcal {K}}} }{\min } \ \underset{\begin{array}{c}{\mathcal {Q}}\subseteq {{\mathcal {D}}_k}, \\ |{\mathcal {Q}}| \ne 0\end{array}}{\min } \ \end{array}\frac{1}{|Y_{{\mathcal {U}},k}||{\mathcal {Q}}|} \log \left(1 + \frac{\sum _{{\mathcal {U}}\in {\mathcal {Q}}}|{\mathbf {h}}_{k}^{H}{\mathbf {v}}_{{\mathcal {U}}}|^2}{\underset{{\mathcal {V}}\in {\mathcal {I}}_k}{\sum }|{\mathbf {h}}_{k}^{H}{\mathbf {v}}_{{\mathcal {V}}}|^2 + N_0}\right)\\\qquad \qquad \mathrm {s.t.} \ \displaystyle \sum _{\begin{array}{c}{\mathcal {U}}\subseteq \hat{{\mathcal {K}}}, |{\mathcal {U}}| = \hat{t}+1\end{array}}\Vert {\mathbf {v}}_{{\mathcal {U}}}\Vert ^2\le P_T,\) 
